Drawing a force diagram and the resultant vector would be extremely helpful for you to realize your mistake. Remember that you can take the arctan of any unitless value, which applies to force/force. Also, keep in mind SOHCAHTOA. It appears you have mistaken sine for cosine in your attempted solution.

Maybe if I point out what you did wrong, you might understand how to proceed.

The angle that you solved for with tan-1(.18/.24) is the angle between the force of q1 on q3 and the force of q2 on q3. This is not the resultant vector. The resultant vector will be the force of q1 on q3 and the force of q2 on q3, added component by component. This is what got you the magnitude, but it will also be what gets you the angle.

Maybe if I point out what you did wrong, you might understand how to proceed.

The angle that you solved for with tan-1(.18/.24) is the angle between the force of q1 on q3 and the force of q2 on q3. This is not the resultant vector. The resultant vector will be the force of q1 on q3 and the force of q2 on q3, added component by component. This is what got you the magnitude, but it will also be what gets you the angle.